Robin Spectrum: Two Disks Maximize the Third Eigenvalue

Alexandre Girouard, Richard S. Laugesen

Research output: Contribution to journalArticlepeer-review


The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided the Robin parameter lies in a certain range and is scaled in each case by the length of the boundary. Equality is achieved when the domain degenerates suitably to the two disks.

Original languageEnglish (US)
Pages (from-to)2711-2742
Number of pages32
JournalIndiana University Mathematics Journal
Issue number6
StatePublished - 2021


  • Neumann
  • Robin
  • Steklov
  • conformal mapping
  • vibrating membrane

ASJC Scopus subject areas

  • General Mathematics


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